A transformation is a change made to the graph of a function, resulting in a new graph.
Transformations can be applied to any function, including exponentialand logarithmicfunctions.
Transformations of Exponential Functions
The general form of an exponential function is:
$$f(x) = a \cdot b^{k(x - d)} + c$$
where:
- $a$ is the vertical stretch or compression factor.
- $b$ is the base of the exponential function.
- $k$ is the horizontal stretch or compression factor.
- $d$ is the horizontal shift.
- $c$ is the vertical shift.
The parent functionof an exponential functionis $f(x) = b^x$.
Vertical Shifts
A vertical shift moves the graph up or down.
- If $c > 0$, the graph shifts up by $c$ units.
- If $c < 0$, the graph shifts down by $|c|$ units.
The parent functionof a logarithmic functionis $f(x) = \log_b(x)$.
Vertical Shifts
A vertical shift moves the graph up or down.
- If $c > 0$, the graph shifts up by $c$ units.
- If $c < 0$, the graph shifts down by $|c|$ units.
1. Sketch the graph of $f(x) = 2^{x+1} - 3$. Describe the transformations applied to the parent function $f(x) = 2^x$. 2. Sketch the graph of $g(x) = \log_3(x - 2) + 1$. Describe the transformations applied to the parent function $g(x) = \log_3(x)$.
How do transformations of functions illustrate the power of mathematical abstraction? Can you think of real-world situations where these transformations might be useful?
